46+47+48+56+65 61+62+63+71+80
- When the T-Number is even, the T-Total is odd.
- When the T-Number is odd, the T-Total is even.
I will now find the rule which links the T-Number with the T-Total.
n+n-9+n-17+n-18+n-19= 5n-63
Example: When n=44 (5x44)-63=157
T-Total=157 25+26+27+35+44=157
T-Totals – 8x8 Grid
T-Number=18 T-Number=37
T-Total=34 T-Total=129
1+2+3+10+18 20+21+22+29+37
T-Number= 58 T-Number=63
T-Total=234 T-Total=259
41+42+43+50+58 46+47+48+55+63
- When the T-Number is odd, the T-Total is odd.
- When the T-Number is even, the T-Total is even.
I will now find the rule which links the T-Number with the T-Total.
n+n-8+n-16+n-17+n-18= 5n-56
Example: When n=36 (5x36)-56=124
T-Total=157 19+20+21+28+36=124
T-Totals – Any sized Grid
I will now find the general rule for any sized grid, which links the T-Number with the T-Total.
n+n-G+n-2G-1+n-2G+n-2G+1= 5n-7G
Example: When n=65, and G=10
T-Number= 65
T-Total= 255 (5x65)-(7x10)
T-Total= 255 44+45+46+55+65
T-Totals - Translating
If I translate the T three vectors right it will become:
T-Number=43 T-Number=46
T-Total=145 T-Total=160
22+23+24+33+43 25+26+27+36+46
- The T-Total has increased by 15.
- This is because there are 5 sequences in the T-Total, which have all increased by 3. 5x3=15.
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Consequently, if the vector is () the formula for moving the T across would be: T-Total+5a
If I translate the T three vectors up it will become:
T-Number=83 T-Number=53
T-Total= 345 T-Total=195
62+63+64+73+83 32+33+34+43+53
- The T-Total has decreased by 150.
- This is because there are 5 sequences in the T-Total, which have all decreased by 3 lots of the Grid number (3G).
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Consequently, if the vector is () the formula for moving the T up would be: T-Total-5bG
- Therefore, if the T moves across by ‘a’ each square increases by ‘a.’
- Therefore, if the T moves up by ‘b’ each square increases by ‘b.’
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To move T by the vector () the formula is
n+n-G+n-2G-1+n-2G+n-2G+1=5n-7G
Consequently, the formula for moving T by any vector is:
n+a-bG+n-G+a-bG+n-2G-1+a-bG+n-2G+a-bG+n-2G+1=5n-7G+5a-5bG
- This shows that if T is translated across by ‘a’ each square increases by ‘a’.
-
This shows that if T is translated up by ‘b’ each square decreases by ‘bG.’
T-Totals – Rotation
I will now find the general rule for rotating T about the point n.
90*
n+n+1+n+2-G+n+2+n+2+G=5n+7
180*
n+n+G+n+2G-1+n+2G+n+2G+1=5n+7G
270*
n+n-1+n-2-G+n-2+n-2+G=5n-7
T-Totals – Rotation about a point
- I will now find the general rule for rotating T about any point. (90*).
n+3-2G-2-3G
replacing the numbers with letters
n+c-dG-d-cG
5n+7
5(n+c-dG-d-cG) +7 5n+5c-5dG-5d-5cG+7
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I will now find the general rule for rotating T about any point. (180*).
n+3-2G-2G+3
replacing the numbers with letters
n+2c-2dG
5n+7G
5(n+2c-2d) +7G 5n+10c-10dG+7G
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I will now find the general rule for rotating T about any point. (270*).
n+3-2G+2+3G
replacing the numbers with letters
n+c-dG+d+cG
5n-7
5(n+c-dG+d+cG) -7 5n+5c-5dG+5d+5cG-7
T-Totals – Combination (Rotation & Translation)
- I will now find the general rule for rotating and then translating a T.
- The rule will only work in certain Grids, because the vector by which I want to translate the T will be too big to fit the limitations of the Grid.
- To do this I must combine the rules for rotation with the ones for translations:
90*= 5n+5c-5dG-5d-5cG+7+5a-5bG
180*= 5n+10c-10dG+7G+5a-5bG
270*= 5n+5c-5dG+5d+5cG-7+5a-5bG
T-Totals –Conclusion
I have found that you can rotate and then translate a T (combination) using the formulae:
90*= 5n+5c-5dG-5d-5cG+7+5a-5bG
This allows you to rotate the T by 90* and then move it by any vector within the Grid limitations.
180*= 5n+10c-10dG+7G+5a-5bG
This allows you to rotate the T by 180* and then move it by any vector within the Grid limitations.
270*= 5n+5c-5dG+5d+5cG-7+5a-5bG
This allows you to rotate the T by 270* and then move it by any vector within the Grid limitations.
Example: Move the T by the vector ()
The new T does not fit within the limitations.
